Page 113 - Textos de Matemática Vol. 34

ORTHOGONAL POLYNOMIALS 103
s = 0,1,2,··· and ν = 0,··· ,N. These polynomials have received much attention in the last decade (see reference [10], where F. Marcell´an and A. Ronveaux gave an updated list of 240 references on this subject).
In proposition in below we state that, in general, if instead of (8.1) we start from a linear combination with any finite number of terms (independent of n) in the left-hand side, and with another one finite number of terms in the right-hand side, then the relation between the moment linear functionals becomes φu = ψv, where φ and ψ are some polynomials which can be explicitly computed. The proof is based in the results presented in the previous sections (see [17]).
Theorem 8.1. Let u and v be two regular functionals in P′, and let {Pn}n≥0 and {Qn}n≥0 be the corresponding MOPS, respectively. Assume that there exist nonnegative integer numbers N and M, and complex numbers ri,n and sk,n (i = 1,··· ,N ; k = 1,··· ,M ; n = 0,1,···), such that the structure relation
NM
Pn(x) + ri,nPn−i(x) = Qn(x) + si,nQn−i(x)
i=1 i=1 holds for all n = 0,1,2,···. Further, assume that
rN,M+N ̸= 0 , sM,M+N ̸= 0 , det[αi,j]N+M ̸= 0, i,j =1
where
0 otherwise ,
with the convention r0,k = s0,ν = 1 for all k = 0,··· ,M and ν = 0,··· ,N. Then
there exist two polynomials φ and ψ, with ∂φ = M and ∂ψ = N, such that φu=ψv.
As a consequence the following proposition holds, which is a slightly modification of Theorem 2.4 in [1].
Corollary 8.2. Let u and v be two regular functionals in P′, and let (Pn)n and (Qn)n be the corresponding MOPS’s, respectively. Then the following two conditions are equivalent:
(i) There exist complex numbers a, b, λ such that (8.3) (x − a)u = λ(x − b)v
and ⟨u,Pn2⟩ ≠ λ⟨v,Q2n⟩ for all n = 1,2,···.
 rj−i,j−1 if 1≤i≤M ∧ i≤j≤N+i,
αij := sj−i+M,j−1 if M+1≤i≤M+N ∧ i−M≤j≤i,
